EXPRESS: People’s Preferences for Different Types of Rational Numbers in Linguistic Contexts

Rational numbers, like fractions, decimals, and percentages, differ in the concepts they prefer to express and the entities they prefer to describe as previously reported in display-rational number notation matching tasks and in math word problem compiling contexts. On the one hand, fractions and percentages are preferentially used to express a relation between two magnitudes, while decimals are preferentially used to represent a magnitude. On the other hand, fractions and decimals tend to be used to describe discrete and continuous entities, respectively. However, it remains unclear whether these reported distinctions can extend to more general linguistic contexts. It also remains unclear which factor, the concept to be expressed (magnitudes vs. relations between magnitudes) or the entity to be described (countable vs. continuous), is more predictive of people’s preferences for rational number notations. To explore these issues, two corpus studies and a number notation preference experiment were administered. The news and conversation corpus studies detected the general pattern of conceptual distinctions across rational number notations as observed in previous studies; the number notation preference experiment found that the concept to be expressed was more predictive of people’s preferences for number notations than the entity to be described. These findings indicate that people’s biased uses of rational numbers are constrained by multiple factors, especially by the type of concepts to be expressed, and more importantly, these biases are not specific to mathematical settings but are generalizable to broader linguistic contexts.

Explicit Values for Ramanujan's Theta Function ϕ(q)

This paper provides a survey of particular values of Ramanujan's theta function $\varphi(q)=\sum_{n=-\infty}^{\infty}q^{n^2}$, when $q=e^{-\pi\sqrt{n}}$, where $n$ is a positive rational number. First, descriptions of the tools used to evaluate theta functions are given. Second, classical values are briefly discussed. Third, certain values due to Ramanujan and later authors are given. Fourth, the methods that are used to determine these values are described. Lastly, an incomplete evaluation found in Ramanujan's lost notebook, but now completed and proved, is discussed with a sketch of its proof.

Factorization invariants of the additive structure of exponential Puiseux semirings

Exponential Puiseux semirings are additive submonoids of [Formula: see text] generated by almost all of the nonnegative powers of a positive rational number, and they are natural generalizations of rational cyclic semirings. In this paper, we investigate some of the factorization invariants of exponential Puiseux semirings and briefly explore the connections of these properties with semigroup-theoretical invariants. Specifically, we provide exact formulas to compute the catenary degrees of these monoids and show that minima and maxima of their sets of distances are always attained at Betti elements. Additionally, we prove that sets of lengths of atomic exponential Puiseux semirings are almost arithmetic progressions with a common bound, while unions of sets of lengths are arithmetic progressions. We conclude by providing various characterizations of the atomic exponential Puiseux semirings with finite omega functions; in particular, we completely describe them in terms of their presentations.

Exploring the Educational Potential of a Game-Based Math Competition

The main aim of this article was to investigate the educational potential of a game-based math game competition to engage students in training rational numbers. Finnish fourth (n = 59; Mage = 10.36) and sixth graders (n = 105; Mage = 12.34) participated in a math game competition relying on intra-classroom cooperation and inter-classroom competition. During a three-week period, the students were allowed to play a digital rational number game, which is founded on number line estimation task mechanics. The results indicated that students benefited significantly from participating in the competition and playing behaviour could be used to assess students rational number knowledge. Moreover, students were engaged in the competition and the results revealed that intrinsically motivating factors such as enjoyment and perceived learning gains predicted students' willingness to participate in math game competitions again. This article provides empirical support that educational game competition can be an effective, engaging, and a fair instructional approach.

ŁOJASIEWICZ INEQUALITY IN P-MINIMAL STRUCTURES

Th purpose of this paper is to extend the Łojasiewicz inequality for functions definable in some subclass of P-minimal structures. More precisely, we prove that the Łojasiewicz inequality holds for functions definable in poptimal expansions of Qp. It is also shown that the Łojasiewicz exponent is a rational number in such p-optimal expansions.

Khintchine’s theorem with rationals coming from neighborhoods in different places

The Duffin–Schaeffer Conjecture answers a question on how well one can approximate irrationals by rational numbers in reduced form (an imposed condition) where the accuracy of the approximation depends on the rational number. It can be viewed as an analogue to Khintchine’s theorem with the added restriction of only allowing rationals in reduced form. Other conditions such as numerator or denominator a prime, a square-free integer, or an element of a particular arithmetic progression, etc. have also been imposed and analogues of Khintchine’s theorem studied. We prove versions of Khintchine’s theorem where the rational numbers are sourced from a ball in some completion of [Formula: see text] (i.e. Euclidean or [Formula: see text]-adic), while the approximations are carried out in a distinct second completion. Finally, by using a mass transference principle for Hausdorff measures, we are able to extend our results to their corresponding analogues with Haar measures replaced by Hausdorff measures, thereby establishing an analogue of Jarník’s theorem.

Improving rational number knowledge using the NanoRoboMath digital game

Rational number knowledge is a crucial feature of primary school mathematics that predicts students’ later mathematics achievement. Many students struggle with the transition from natural number to rational number reasoning, so novel pedagogical approaches to support the development of rational number knowledge are valuable to mathematics educators worldwide. Digital game-based learning environments may support a wide range of mathematics skills. NanoRoboMath, a digital game-based learning environment, was developed to enhance students’ conceptual and adaptive rational number knowledge. In this paper, we tested the effectiveness of a preliminary version of the game with fifth and sixth grade primary school students (N = 195) using a quasi-experimental design. A small positive effect of playing the NanoRoboMath game on students’ rational number conceptual knowledge was observed. Students’ overall game performance was related to learning outcomes concerning their adaptive rational number knowledge and understanding of rational number representations and operations.

Powers of Elliptic Scator Numbers

Elliptic scator algebra is possible in 1+n dimensions, n∈N. It is isomorphic to complex algebra in 1 + 1 dimensions, when the real part and any one hypercomplex component are considered. It is endowed with two representations: an additive one, where the scator components are represented as a sum; and a polar representation, where the scator components are represented as products of exponentials. Within the scator framework, De Moivre’s formula is generalized to 1+n dimensions in the so called Victoria equation. This novel formula is then used to obtain compact expressions for the integer powers of scator elements. A scator in S1+n can be factored into a product of n scators that are geometrically represented as its projections onto n two dimensional planes. A geometric interpretation of scator multiplication in terms of rotations with respect to the scalar axis is expounded. The powers of scators, when the ratio of their director components is a rational number, lie on closed curves. For 1 + 2 dimensional scators, twisted curves in a three dimensional space are obtained. Collecting previous results, it is possible to evaluate the exponential of a scator element in 1 + 2 dimensions.